[Inquiry] Re: Introduction to Inquiry Driven Systems

[Jon Awbrey]

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INT.  Note 26

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2.2.3.2.  Transfer

What is it that gives a distinctively inductive character
to the acquisition of a knowledge base?  It is evidently the
"analogy of experience" that underlies the useful application.
Whenever we find ourselves prefacing an argument with the phrase
"If past experience is any guide ..." then we can be sure that
this principle has come into play.  We are invoking an analogy
between past experience, considered as a totality, and present
experience, considered as a point of application.  What we mean
in practice is this:  "If past experience is a fair sample of
possible experience, then the knowledge gained in it applies
to present experience".  This is the mechanism that allows a
knowledge base to be carried across gulfs of experience that
are indifferent to the effective contents of its rules.

Here are the details of how this notion of transfer
works out in the case of the "Sign of Rain" example:

Let us consider a fragment, K_pres, of the reasoner's
knowledge base that is logically equivalent to the
conjunction of two rules:

   K_pres  =  (B => A) and (B => D).

K_pres = present knowledge base, expressed in the form of
a logical constraint on the present universe of discourse.

It is convenient to have the option of expressing all logical statements
in terms of their models, that is, in terms of the primitive circumstances
or the elements of experience over which they hold true.

1.  Let E_past be the chosen set of experiences,
    or the circumstances that we have in mind
    when we refer to "past experience".

2.  Let E_poss be the collective set of experiences,
    or the projective total of possible circumstances.

3.  Let E_pres be the present experience,
    or the circumstances that are present
    to the reasoner at the current moment.

If we think of the knowledge base K_pres as referring
to the "regime of experience" over which it is valid,
then all of these sets of models can be compared by the
simple relations of set inclusion or logical implication.

Figure 4 schematizes this way of viewing the "analogy of experience".

o-----------------------------------------------------------o
|                                                           |
|                          K_pres                           |
|                             o                             |
|                            /|\                            |
|                           / | \                           |
|                          /  |  \                          |
|                         /   |   \                         |
|                        /  Rule   \                        |
|                       /     |     \                       |
|                      /      |      \                      |
|                     /       |       \                     |
|                    /     E_poss      \                    |
|              Fact /         o         \ Fact              |
|                  /        *   *        \                  |
|                 /       *       *       \                 |
|                /      *           *      \                |
|               /     *               *     \               |
|              /    *                   *    \              |
|             /   *  Case           Case  *   \             |
|            /  *                           *  \            |
|           / *                               * \           |
|          /*                                   *\          |
|         o<<<---------------<<<---------------<<<o         |
|      E_past         Analogy Morphism         E_pres       |
|    More Known                              Less Known     |
|                                                           |
o-----------------------------------------------------------o
Figure 4.  Analogy of Experience

In these terms, the "analogy of experience" proceeds by inducing a Rule
about the validity of a current knowledge base and then deducing a Fact,
its applicability to a current experience, as in the following sequence:

Inductive Phase:

   Given  Case:  E_past => E_poss, Chosen events fairly sample Collective events.
   Given  Fact:  E_past => K_pres, Chosen events support the Knowledge regime.
   Induce Rule:  E_poss => K_pres, Collective events support the Knowledge regime.

Deductive Phase:

   Given  Case:  E_pres => E_poss, Current events fairly sample Collective events.
   Given  Rule:  E_poss => K_pres, Collective events support the Knowledge regime.
   Deduce Fact:  E_pres => K_pres, Current events support the Knowledge regime.

Jon Awbrey

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